MA2223: PROBLEM SHEET 1 is exactly 2r.

1. Show that the diameter of an open ball B(a, r) in Euclidean space
Rn is exactly 2r.
2. Let C[0, 1] be the set of continuous real-valued functions on the
interval [0, 1] together with the supremum metric
d(f, g) = sup |f (x) − g(x)|
Let f ∈ C[0, 1] be the function f (x) = x. Sketch the graph of f and
the region which contains the graphs of all elements of the open ball
B(f, 1).
3. Let (X, d) be a metric space and let A be a subset of X. The set of
all interior points of A is called the interior of A and is denoted A◦ .
(a) Prove that A◦ is an open set in (X, d).
(b) Prove that for any subset B we have (A ∩ B)◦ = A◦ ∩ B ◦ .
4. Let X be a non-empty set and let d : X × X → R satisfy the
following conditions
(a) d(x, y) = 0 if and only if x = y,
(b) d(x, z) ≤ d(y, x) + d(z, y) for all x, y, z ∈ X.
Show that d is a metric on X.